Holomorphic separability

In mathematics in complex analysis, the concept of holomorphic separability is a measure of the richness of the set of holomorphic functions on a complex manifold or complex-analytic space.

Formal definition

A complex manifold or complex space ${\displaystyle X}$ is said to be holomorphically separable, if whenever x ≠ y are two points in ${\displaystyle X}$, there exists a holomorphic function ${\displaystyle f\in {\mathcal {O}}(X)}$, such that f(x) ≠ f(y).^[1]

Often one says the holomorphic functions separate points.

Usage and examples

• All complex manifolds that can be mapped injectively into some ${\displaystyle \mathbb {C} ^{n}}$ are holomorphically separable, in particular, all domains in ${\displaystyle \mathbb {C} ^{n}}$ and all Stein manifolds.
• A holomorphically separable complex manifold is not compact unless it is discrete and finite.
• The condition is part of the definition of a Stein manifold.